Additivity of maps on triangular algebras.
In 1950 N. Jacobson proved that if u is an element of a ring with unit such that u has more than one right inverse, then it has infinitely many right inverses. He also mentioned that I. Kaplansky proved this in another way. Recently, K. P. Shum and Y. Q. Gao gave a new (non-constructive) proof of the Kaplansky-Jacobson theorem for monoids admitting a ring structure. We generalize that theorem to monoids without any ring structure and we show the consequences of the generalized Kaplansky-Jacobson...
This paper contains the algebraic analog for idempotent matrices of the Chern-Weil theory of characteristic classes. This is used to show, algebraically, that the canonical line bundle on the complex projective space is not stably trivial. Also a theorem is proved saying that for any smooth manifold there is a canonical epimorphism from the even dimensional algebraic de Rham cohomology of its algebra of smooth functions onto the standard even dimensional de Rham cohomology of the manifold.